Development of a Deeply-Coupled GPS/INS Integration Algorithm using Quaternions

Yuhong Yang, Junchuan Zhou, Holger Nies, Otmar Loffeld

Center for Sensor Systems (ZESS) University of Siegen

Siegen, Germany

Stefan Knedlik iMAR GmbH

Im Reihersbruch 3 St. Ingbert, Germany

Abstract—A deeply-coupled Global Positioning System (GPS)/ Inertial Navigation System (INS) integration algorithm is proposed in this paper. The mathematical system process and observation models are provided. Due to the nonlinearity of the system models, an Extended Kalman Filter (EKF) is employed, which uses quaternions as the representation of attitude. The integrated algorithm is tested using the IFEN GPS radio frequency (RF) signal simulator. Both static and dynamic scenarios are simulated. Numerical results are compared and analyzed.

Keywords—Deeply-coupled; GPS/INS; Quaternions

I. INTRODUCTION

Deeply-coupled GPS/INS integration research has drawn more attention in the recent few years. Traditional GPS/INS integration employs a loosely or a tightly-coupled architecture. One example of different implementations of GPS/INS integration systems is illustrated in Fig.1.

RF Front End AcquisitionTracking loops(DLL, FLL/PLL)

Pseudorange & Delta Range

Navigation Filter

Strap-downProcessing

IMU(3 Gyro + 3 Acc)

Integration

Kalman

Filter

Navigation Solution

IMU

GNSS Receiver Corrections

IntegrationAlgorithm

Measurements

③ DeeplyDiscriminatorOutput Based on Is and Qs

② TightlyPR., DPR.

① LooselyP. ,V.

P. V. tP. V. t

Fig. 1. GPS/INS integration architectures.

The loosely-coupled integration has a decentralized estimation architecture, which uses the output information of the navigation solutions from a GPS receiver and an INS. The main advantages of using loosely-coupled integration are: a) the system observation model is simpler, and accordingly it requires much less computational burden; b) the number of measurement inputs (i.e., position and velocity) for the Kalman filter is fixed; c) redundant GPS navigation solutions are available. However, the disadvantages are: a) in case of using two separate KFs (i.e., one for GPS navigation processing, and the other for integration purpose), it opens the possibility of presenting instable navigation solutions caused by mutual feedbacks of estimation errors, which is coined as cascaded filtering problem; b) usually more than 4 satellites are required to maintain a GPS navigation solution.

Unlike the loosely-coupled manner, in a tightly-coupled integration, only a centralized KF is used. The pseudorange and delta range (or Doppler) measurements are directly used in the filter. The advantages of applying the tightly-coupled integration are mainly in the following aspects: a) the cascaded filtering problem arising through the mutual feedbacks of the estimation errors between two separate KFs is eliminated; b) the system does not require a GPS navigation solution to aid the INS. Therefore, even with less than 4 satellites in view, the remaining satellite based measurements can still be used in the algorithm, which promotes the robustness of the navigation system; c) all systematic errors and noise sources of the distributed sensors are modeled in the same filter, which ensures that all error correlations are accounted for. Nevertheless, the disadvantage of tightly-coupled approach arises in the increased dimension of the observation vector with respect to the loosely-coupled manner. And the number of GPS measurements as inputs to the Kalman filter is varying in practical applications, which depends on the signal environments.

It is known that in a tightly-coupled system, all GPS tracking channels are working independently. In this work, we move one step forward to implement the deeply-coupled integration. The algorithm we proposed uses quaternions as the representation of attitude. This work is based on a software-defined GPS receiver we developed. The former work on this topic can be found in [1-3]. In this system, an EKF is employed to fuse the outputs (i.e., code phase error and frequency error) from the receiver tracking loops and Inertial Measurement Unit (IMU) observables to enhance GPS tracking performance, which gives robust and smoothed navigation results.

II. IMPLEMENTATION OF A DEEPLY-COUPLED

INTEGRATION SYSTEM

A. System Architecture

The realization of a deeply-coupled integration can be mainly divided into two categories [4]. One way directly uses the GPS receiver’s baseband correlator outputs (i.e., Is and Qs) as the measurements to fuse with the output of IMU. The other way is based on the design of a vector tracking loop, which was first proposed by Spilker [5]. He used nonlinear discriminator outputs as measurements. In this paper, our development is based on the second approach. The high level architecture of the algorithm is illustrated in Fig. 2. We employ an IMU, which contains 3-axis gyroscope and 3-axis accelerometer.

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16th International Conference on Information Fusion Istanbul, Turkey, July 9-12, 2013

978-605-86311-1-3 ©2013 ISIF

RF Front End

Correlator Discriminator

IntegrationKalman

Filter

Transformed to Code phase and

carrier frequency

NCOCode/Carrier

Generator

Strap-downProcessing

IMU(3 Gyro + 3 Acc)

I, QCode Phase errors &Frequency errors

P.,V.,t

Corrections

Measurements

Acquisition

Fig. 2. The architecture of the deeply-coupled system.

In Fig.2, the acquisition block (at the left side of the plot) is used to acquire the incoming signals from the available satellites, and to provide code phase and coarse Doppler frequency to the tracking loop. In the tracking loop, the carrier frequency is refined to several Hz precision. Unlike the traditional GPS receiver, our deeply-coupled system uses a global integration Kalman filter for all channels. For this reason, the channels are no longer working independently, but are coupled to a common navigation solution. Thus, with the assistance from the IMU, the channels can principally help each other in the case of GPS signal attenuation conditions.

In such a deeply-coupled system, the essential relationships are the connection between the code phase error and the position, clock error, and the connection between the carrier frequency error and velocity, clock drift. They have the mathematical relations as shown in (1). Details can be also found in [6].

, , , ,

, , , ,

ˆ

ˆ

Tj k j k k b k j k

f T fj k j k k d k j k

z LOS t w

z LOS t w

p

v

(1)

where ,j kz and ,

fj kz are the code phase error [m] and carrier

frequency errors [m/s] at epoch k for certain tracking loops; ˆ

kp and ˆ k v are the estimated position error and velocity

error at epoch k; ,j kLOS represents the unit line of sight

vector from the receiver to the j-th satellite; ,b kt is the

receiver clock bias in distance, and ,d kt denotes the clock

drift error [m/s]; ,j kw and ,fj kw represent the white Gaussian

noise errors.

B. INS Principle

Inertial navigation is based on Newtonian physics and is affected by gravity. That is, the object will remain in uniform motion unless disturbed by an external force. It involves a blend of inertial measurements, mathematics, control system design and geodesy [7]. The external force generates acceleration on the object, which can be measured by the inertial sensor. After the integration of the measured

accelerations, under consideration of measurements from gyroscopes, the change in velocity and position with respect to the initial conditions can be determined. A conventional IMU consists of three gyroscopes for measuring angular rates and three accelerometers for measuring accelerations. They are mounted in triads so that the sensitive axes of sensors are mutually orthogonal, setting up a Cartesian reference frame. For a strapdown inertial sensor platform, the inertial sensor is rigidly mounted to the structure of the vehicle. The sensor raw data are processed to yield navigation solutions (i.e., position, velocity, and attitude over time), and this process is named strapdown processing. The INS strapdown processing models in continuous and discrete time domains are given in the next section using quaternions as the representation of attitude.

C. INS Strapdown and System Process Models

We will not use Euler angles as representation of attitude due to the inherent problem of singularity [7]. In this paper, the quaternion vector is denoted as 1[ ]T Tq q q

, where

2 3 4[ ]Tq q q q

. It is used to represent the rotation from the

navigation frame to the body frame. The attitude differential equation in terms of the

quaternion vector q is given in (2). For a detailed derivation,

the reader is referred to [8, 9].

01,

2 [ ]b bbn bn

Tbbn

b bbn bn

with

ω ω

ωq Q q Q

ω ω

(2)

where b

bnω is the angular rate measurement vector from the

navigation frame to the body frame, expressed in the body frame, which is equal to:

b b bbn in ib ω ω ω (3)

where b

ibω is the rotational rate vector of the body frame

relative to the inertial frame, expressed in the body frame (i.e., IMU gyroscope raw measurements); b

inω represents the sum

of the rotation of the Earth with respect to the inertial frame plus the turn rate of the navigation frame with respect to the

Earth, expressed in the body frame, i.e., b b n nin n ie en ω R ω ω .

Using a low-cost MEMS-based IMU, the Earth rotation is usually buried in sensor errors, and cannot be detected. The Coriolis and centrifugal terms are not considered in the following sections. Moreover, for short distance applications, the transport rate is negligible. Considering these effects, we have b

in ω 0 . Thus, (3) turns to be:

b b b bbn ib bn ib ω 0 ω ω ω (4)

where , , ,

Tb b b bib ib x ib y ib z ω is the gyroscope raw data,

resolved in the body frame.

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Using (3) and (4), (2) can be approximated as:

1

2bbn

ω

q Q q ,

with 0 0

[ ] [ ]bbn

T Tb bbn ib

b b b bbn bn ib ib

ω

ω ωQ

ω ω ω ω

(5)

The simplified strapdown mechanization model for the

IMU can be expressed in the navigation frame, as shown in (6)

( )

1

2bbn

n n

n bn b ib n

ω

p v

v R q f g

q Q q

(6)

where np stands for position and nv is velocity in navigation

frame; ng represents gravity indicated in the navigation

frame, which is assumed to be approximately constant in the region of interest.

The rotational transformation matrix ( )nbR q from the body

frame to the navigation frame is expressed using quaternions:

2 2 2 21 2 3 4 2 3 1 4 1 3 2 4

2 2 2 22 3 1 4 1 2 3 4 3 4 1 2

2 2 2 22 4 1 3 1 2 3 4 1 2 3 4

2( ) 2( )

( ) 2( ) 2( )

2( ) 2( )

T

nb

q q q q q q q q q q q q

q q q q q q q q q q q q

q q q q q q q q q q q q

R q

(7)

The IMU specific force and angular rate measurement errors (e.g., sensor biases) are modeled as constants superseded by random walk, which are given in (8):

biasb f

biasb

f w

ω w

(8)

where fw and w are assumed to be zero mean, Gaussian

distributed. It is worth mentioning that using (6) as the GPS/INS

system propagation model, the IMU incoming measurements should be compensated by the current estimate of the sensor

biases before further processing, i.e., b b biasib ib b f f f f n ,

b b biasib ib b w ω ω ω n ( fn and wn are the remaining noises

which are assumed to be zero-mean, Gaussian distributed). In the discrete time domain with a sufficiently small time

interval (e.g., 0.01s IMU update time), and for low dynamic applications, we have:

,

, 1 , ,

, 1 , ,

1

, 1 , ,

, 1 , ,

( )

1

2bbn k

n k n k n k

n bn k n k b k ib k n

k k k

bias biasb k b k f k

bias biasb k b k k

T

T

T

ω

p p v

v v R q f g

q q Q q

f f w

ω ω w

(9)

where “T” is system propagation time interval.

In (9), a part of the model is nonlinear, e.g., ( )n

bR q

contains quadratic terms of quaternion elements. Therefore, the linearization process should be conducted in order to apply Kalman filtering. An EKF is used in this paper, which is based on a first order linearization process on the stochastic system model with the assumption of Gaussian distributed noises. Besides, in the scope of a deeply-coupled integration approach, the receiver clock errors need to be modeled. The range-rate equivalent of the clock drift error is modeled as a constant plus a random walk process, while the range equivalent of the receiver clock bias error is the integral of the clock drift error. Thus, the linearized system propagation model used for GPS/INS deeply-coupled integration is formulated in (10) at the bottom of the page, where error states are employed.

,

3 3 3 3 3 4 3 3 3 3 3 1 3 1, 1

3 3 3 3 23, 3 3 3 1 3 1, 1

1 4 3 4 3 4 4 4 3 4 1 4 1

, 13 3 3 3 3 4

, 1

1

1

( )

1 1

2 2b

kbn k

n kn

k b kn k

kbiasb k

biasb k

k

k

T

T T

T T

c t

c t

qω

I I O O O 0 0pO I F R q O 0 0v

q O O I Q O Ω 0 0f

O O O Iω

,

,

,

3 3 3 3 3 1 3 1,

3 3 3 3 3 4 3 3 3 3 3 1 3 1

,3 1 3 1 4 1 3 1 3 1

,3 1 3 1 4 1 3 1 3 1

1

0 1

n k

n k

kbiasb k kbiasb k

T T T T T b k

T T T T T b k

c tT

c t

p

v

q

f wO 0 0

ωO O O O I 0 0

0 0 0 0 0

0 0 0 0 0

(10)

where

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2 3 4

1 4 3

1 3 3 4 1 2

3 2 1

k

T

k

k

q q q

q q q

q q q q

q q q

q

qΩ

I q

(11)

and 23,kF is computed as:

1 4 3 2 3 4 3 2 1 1 4 2

23, 1 4 2 3 2 1 2 3 4 1 4 3

3 2 1 1 4 2 1 4 3 2

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ2

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

x y z x y z x y z y x z

k y x z x y z x y z x y z

x y z y x z x y z

q f q f q f q f q f q f q f q f q f q f q f q f

q f q f q f q f q f q f q f q f q f q f q f q f

q f q f q f q f q f q f q f q f q f q

F

3 4ˆ ˆ ˆx y z

kf q f q f

(12)

In (12), , ,ˆ ˆb biasx ib x b x

kf f f , , ,

ˆ ˆb biasy ib y b y

kf f f and

, ,ˆ ˆb biasz ib z b z

kf f f are the IMU raw data compensated by the

current estimate of the sensor biases expressed in the body frame.

D. System Measurement Model

For the measurement update, we have

1, 1, 3 1

, , 3 1

, 11, 3 1 1,

, 1

, 3 1 , 2 8

1 0

1 0

0 1

0 1

T Tk k

k

T Tkj k j k

kf T Tb kk k

d k

f T Tj k j k j

z LOS

z LOS

tz LOS

t

z LOS

0

p

v0w

0

0

(13)

where ,j kz and ,

fj kz are code phase and carrier frequency

discriminator error outputs. Equation (13) is the matrix representation of (1).

For the code phase error, we employ a delay lock loop (DLL) discriminator, which is a non-coherent early minus late envelope normalized by E+L, to remove the amplitude sensitivity [10].

1

2

E L

E L

(14)

where 2 2 2 2,E E L LE I Q L I Q . EI and LI represent sum

of the in-phase early and late correlator outputs respectively between time epoch k and k+1; EQ and LQ represent the sum of quadrature early and late correlation results between k and k+1 epoch.

For the carrier frequency error, we employ a normalized decision directed frequency lock loop (FLL) discriminator to eliminate the data bit transition sensitivity [10].

2 1

( )

2 ( )

cross sign dot

t t

(15)

where 1 2 1 2P P P Pdot I I Q Q , 1 2 2 1P P P Pcross I Q I Q .

The 1PI and 2PI denote the in-phase prompt accumulated

correlation results over time epoch k-1 to k and k to k+1; 1PQ

and 2PQ denote the quadrature prompt correlation results over

time periods.

III. EXPERIMENTS

A. Experiment Setup

The experiment setup is shown in Fig. 3. The simulation is conducted using the trajectories generated from the IFEN hardware GPS RF signal simulator NavX®-NCS. The high frequency GPS signals are sampled, down converted and collected by the hardware front end GN3Sv3 from Sparkfun at the sampling frequency of 38.184 kHz. Then the intermediate frequency (IF) signals are handled over to the integration algorithm as the GPS signal input for post processing. The IMU data is generated using INS Toolbox 2.0 (2005) for Matlab developed by GPSoft Ltd. The free Flight Dynamic Control Toolbox is used to calculate the initial attitudes of the platform based on the simulated dynamic trajectory.

Fig. 3. Simulation experiment setup.

The IMU data is simulated based on the parameters from a LandmarkTM 20 eXT MEMS-IMU. The main specification parameters are given in Table 1.

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TABLE I. SPECIFICATION OF LANDMARKTM 20 EXT MEMS-IMU

Gyroscope (Angular rates)

Bias in-run Stability (1 )

Noise(ARW) (1 )

Scale factor error [ppm]

20 [°/h] 0.035[°/s/√Hz] ≤1000Accelerometer

(Specific forces) Bias in-run

Stability (1 ) Noise(ARW)

(1 ) Scale factor error [ppm]

20 [μg] 40 [μg/√Hz] ≤1000

B. Straight Line Path

In the first experiment, a straight line trajectory is generated by the signal simulator with constant velocity of 110 m/s toward east. For comparision puporse, red curve of Fig. 4 is the navigation results from a deeply-coupled sytem. The blue dots are the standalone GPS solution using the least-squares estimator.

4850

4900

4950

5000

5050

5100

5150

5200

5250

−20−10010

−20

0

20

40

East [m]

North [m]

Up

[m]

scalar tracking loop + least−squaresINS−aided tracking loop

Fig. 4. 3-D position navigation results. As shown in Fig. 4, the plot is given in ENU (east-north-

up) frame. The deeply-coupled system presents much better navigation solution than the standalone GPS solution. This is because that the IMU estimates can smooth the noisy code phase and frequency errors, which stabilizes the tracking loop. The prompt in-phase components from 6 tracking loops are plotted in Fig. 5 to show the internal working of the tracking.

0 1000 2000 3000 4000−1

0

1x 10

4 PRN 26 (Ip)

Epoch [1ms]0 1000 2000 3000 4000

−1

0

1x 10

4 PRN 17 (Ip)

Epoch [1ms]

0 1000 2000 3000 4000−1

0

1x 10

4 PRN 22 (Ip)

Epoch [1ms]

0 1000 2000 3000 4000−1

0

1x 10

4 PRN 27 (Ip)

Epoch [1ms]0 1000 2000 3000 4000

−1

0

1x 10

4 PRN 02 (Ip)

Epoch [1ms]

0 1000 2000 3000 4000−1

0

1x 10

4 PRN 06 (Ip)

Epoch [1ms]

Fig.5. In phase component of tracking loops in a deeply-coupled system.

Fig. 5 shows that with the aiding from IMU observable, the tracking loops work fine. In this way, all the channels are connected to each other by the navigation filter, and each channel is able to correctly track the GPS satellite signals and decode the navigation data.

The similar smoothed behaviour can also be observed in the velocity solution. Fig. 6 shows that the velocity of the object is also correctly estimated. The estimated east velocity is about 110 m/s, where in the north and up directions, the estimates are around zero.

0 500 1000 1500 2000 2500 3000 3500 4000109

109.5

110

110.5

111

Time [ms]

Eas

t [m

/s]

scalar tracking loop + least−squaresINS−aided tracking loop

0 500 1000 1500 2000 2500 3000 3500 4000−1

−0.5

0

0.5

1

Time [ms]N

orth

[m/s

]

0 500 1000 1500 2000 2500 3000 3500 4000−2

−1

0

1

2

Time [ms]

Up

[m/s

]

Fig. 6. Velocity estimation comparison in ENU navigation frame.

The results of straight line test show that the deeply-coupled integration system gives smoothed results in constant acceleration scenario. This is because a Kalman filter is used inside which fuses information from all the channels by estimating the common state, and the inertial sensors can provide dynamic measurements (in this test, they are zeros plus sensor errors), which are used in the system process model. Practically, using true dynamic measurements turns out to be better than using dynamic models. Because in reality, the true dynamic of the platform may often not be described by any model due to the lack of enough a priori knowledge on the trajectory.

For testing the algorithm with changing accelerations, a clockwise circle path is employed in the next section.

C. Clockwise Circle Path

In this test, a clockwise circle path is simulated. The platform flies with constant norm velocity of 30 m/s. The path radius is 1 km, with circle center point 48.1815° (latitude), 11.808° (longitude), 525 m (height). In this case, in the horizontal plane, the accelerations (i.e., the output of x and y-axis accelerometers) are changing over time. For the attitude, the heading (i.e., the output of gyro-z axis) is changing constantly over time. The position and velocity navigation results for 20 seconds are compared with GPS alone solution

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(using a least-squares estimator) in Fig. 7 and Fig. 8. In both figures, the red curves are the results from the deeply-coupled integration, while the blue curves are the results from GPS stand-alone system. The figures show that with INS aiding, the integration solution is less noisy and much smoother as compared with a GPS only solution.

0 5.000 10.000 15.000 20.000−600

−500

−400

−300

−200

−100

0

100

Time [ms]

Eas

t [m

]

0 5.000 10.000 15.000 20.000−50

0

50

100

150

200

250

300

Time [ms]

Nor

th [m

]

0 5.000 10.000 15.000 20.000−40

−20

0

20

40

60

Time [ms]

Up

[m]

−500

−400

−300

−200

−100

0

050100150200250

−200

2040

East [m]

North [m]

Up

[m]

scalar tracking loop + least−squaresINS−aided tracking loop

Fig.7. Navigation solution comparisons in ENU navigation frame.

0 2.000 4.000 6.000 8.000 10.000 12.000 14.000 16.000 18.000 20.000−35

−30

−25

−20

Time [ms]

Eas

t [m

/s]

scalar tracking loop + least−squaresINS−aided tracking loop

0 2.000 4.000 6.000 8.000 10.000 12.000 14.000 16.000 18.000 20.0000

5

10

15

20

25

Time [ms]

Nor

th [m

/s]

0 2.000 4.000 6.000 8.000 10.000 12.000 14.000 16.000 18.000 20.000−5

0

5

Time [ms]

Up

[m/s

]

Fig. 8. Velocity estimation comparison in ENU navigation frame.

In this test, the advantage of the deeply-coupled system comes from the fact that the IMU can measure the true dynamics of the platform, and this information is used in the tracking loops which can improve the tracking behavior. Besides, in the case that the platform encounters challenging signal environments (e.g., GPS signals are attenuated or blocked), the system will show its outperformance by using the IMU estimates to smooth and bridge the outage periods, which will also speed up the reacquisition of signals when the vehicle

leaves the outage environments. The proof of these benefits goes to the future work.

IV. CONCLUSION

In this paper, we have presented an implementation of an algorithm for a deeply-coupled GPS/INS system. The system models, using quaternions as the representation of attitude, are given. An extended Kalman filter is employed as the tool to fuse the data from the GPS receiver tracking loop outputs and IMU observables. In our system, the separated tracking channels are connected with each other due to the fact that global navigation filter estimates the common states. In this way, the channels ‘can help each other’ in the tracking which is a big difference from the tightly-coupled system. Besides, the IMU can capture the dynamics of the platform to improve the navigation solution for example in challenging GPS signal environment, and to enhance the system robustness. Using even a cheap IMU is better than using dynamic models in the Kalman filter, as in reality. The dynamics of the platform can often not be accurately described by any models. Two simulations have been made to verify the approach.

ACKNOWLEDGMENT

The first author gratefully acknowledges the support provided by the programme -Multi Modal Sensor Systems for Environmental Exploration and Safety (MOSES) within the Center for Sensor Systems (ZESS), University of Siegen, Germany.

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